Square-free decomposition is one of fundamental computations for polynomials. However, any conventional algorithm may not work for polynomials with a priori errors on their coefficients. There are mainly two approaches to overcome this empirical situation: approximate polynomial GCD (greatest common divisor) and the nearest singular polynomial. In this paper, we show that these known approaches are not enough for detecting the nearest square-free part (which has no multiple roots) within the given upper bound of perturbations (a priori errors), and we propose a new definition and a new method to detect a square-free part and its decomposition numerically by following a recent framework of approximate polynomial GCD.

无平方分解是多项式的基本计算之一。然而，任何常规算法可能不适用于在其系数上具有先验误差的多项式。主要有两种方法可以克服这种经验情况：近似多项式GCD（最大公约数）和最接近的奇异多项式。在本文中，我们证明了这些已知方法不足以检测给定摄动上限（先验误差）内最近的无平方部分（无多个根），并提出了新的定义和新的遵循最近的多项式GCD框架，用数字方法检测无平方零件及其分解的数值。